Systems and methods to obtain a myocardial mass index indicative of an at-risk myocardial region

ABSTRACT

Systems and methods to obtain a myocardial mass index indicative of an at-risk myocardial region. In at least one embodiment, a method for diagnosing a risk of cardiac disease is provided, the method comprising the steps of identifying a luminal cross-sectional area of a side branch vessel, identifying a luminal cross-sectional area of a main artery most proximal to the side branch vessel, determining a myocardial region at risk for infarct from side branch occlusion relative to a mass perfused by the most proximal artery, wherein such region is based on a relationship between the cross-sectional areas of the side branch and the most proximal main artery, and diagnosing a risk of cardiac disease based on a diameter of the side branch and the size of the myocardial region at risk for infarct perfused by the side branch.

PRIORITY

The present U.S. continuation-in-part application is related to, andclaims the priority benefit of, U.S. patent application Ser. No.12/864,016, filed Jul. 22, 2010, which is related to, claims thepriority benefit of, and is a U.S. §371 national stage patentapplication of, International Patent Application Serial No.PCT/US08/72925, filed Aug. 12, 2008, which is related to, claims thepriority benefit of, and is an international continuation-in-partapplication of, International Patent Application Serial No.PCT/US08/00762, filed Jan. 22, 2008, which is related to, and claims thepriority benefit of, U.S. Provisional Patent Application Ser. No.60/881,833, filed Jan. 23, 2007. The contents of each of theseapplications are hereby incorporated by reference in their entirety intothis disclosure.

BACKGROUND

The disclosure of the present application relates generally to diagnosisof vascular disease and cardiovascular disease, in particular relatingto using morphological features of the coronary artery tree to diagnosecoronary artery disease and the risk of future cardiac events.

Diffuse coronary artery disease (DCAD), a common form ofatherosclerosis, is difficult to diagnose because the arterial lumencross-sectional area is diffusely reduced along the length of thevessels. Typically, for patients with even mild segmental stenosis, thelumen cross-sectional area is diffusely reduced by 30 to 50%. Thefailure of improved coronary flow reserve after angioplasty may mainlybe due to the coexistence of diffuse narrowing and focal stenosis.Whereas angiography has been regarded as the “gold standard” in theassessment of focal stenosis of coronary arteries, its viability todiagnose DCAD remains questionable. The rationale of conventionalangiography in the assessment of coronary artery disease is to calculatethe percent lumen diameter reduction by comparison of the target segmentwith the adjacent ‘normal’ reference segment. In the presence of DCAD,however, an entire vessel may be diffusely narrowed so that no truereference (normal) segment exists. Therefore, in the presence of DCAD,standard angiography significantly underestimates the severity of thedisease.

To overcome the difficulty of using angiography in the diagnosis ofDCAD, intravascular ultrasound (IVUS) has been the subject of extensivestudies. IVUS has the advantage of directly imaging the cross-sectionalarea along the length of the vessel using a small catheter. Thedisadvantage of IVUS, however, is that its extensive interrogation ofdiseased segments may pose a risk for plaque rupture.

In addition to the foregoing, biological transport structures (vasculartrees, for example), have significant similarities despite remarkablediversity and size across species. The vascular tree, whose function isto transport fluid within an organism, is a major distribution system,which has known fractal and scaling characteristics. A fundamentalfunctional parameter of a vessel segment or a tree is the hydraulicresistance to flow, which determines the transport efficiency. It isimportant to understand the hydraulic resistance of a vascular treebecause it is the major determinant of transport in biology.

In a hydrodynamic analysis of mammalian and plant vascular networks, amathematical model of ¾-power scaling for metabolic rates has beenreported. A number of scaling relations of structure-function featureswere further proposed for body size, temperature, species abundance,body growth, and so on. Although the ¾ scaling law was originallyderived through a hemodynamic analysis in the vascular tree system, atleast one basic structure-function scaling feature of vascular treesremains unclear: “How does the resistance of a vessel branch scale withthe equivalent resistance of the corresponding distal tree?”

What is needed is an improved approach to diagnosis and prognosis ofvascular disease and cardiovascular disease and its symptoms that avoidintrusive and expensive methods while improving accuracy and efficacy.Such an approach may include, for example, a novel scaling law of asingle vessel resistance as relative to its corresponding distal tree,or scaling of myocardial mass to vessel caliber.

Blood pressure and perfusion of an organ depend on a complex interplaybetween cardiac output, intravascular volume, and vasomotor tone,amongst others. The vascular system provides the basic architecture totransport the fluids while other physical, physiological, and chemicalfactors affect the intravascular volume to regulate the flow in thebody. Although the intravascular volume can adapt to normal physicaltraining, many diagnostic and treatment options depend on the estimationof the volume status of patients. For example, a recent study classifiedblood volume status as hypovolemic, normovolemic, and hypervolemic.

Heart failure results in an increase of intravascular volume(hypervolemia) in response to decreased cardiac output and renalhypoperfusion. Conversely, myocardial ischemia and infarct lead to adecrease of intravascular volume (hypovolemia) distal to an occludedcoronary artery, and patients with postural tachycardia syndrome alsoshow hypervolemia. Furthermore, patients of edematous disorders havebeen found to have abnormal blood volume. Currently, there is nononinvasive method to determine the blood volume in sub-organ, organs,organ system or organism. The disclosure of the present applicationprovides a novel scaling law that provides the basis for determinationof blood volume throughout the vasculature.

In addition to the foregoing, the reduction of blood flow to the heartdue to occlusion of a side branch (SB) vessel may cause elevation ofbiomarkers after coronary stenting of a bifurcation, as well as apotential increase in future risk of cardiac events (includingmortality). Hence, it is important to understand the relation betweenthe lumen caliber of the SB and the myocardial mass it nourishes todetermine the portion of myocardial tissue at risk for infarct if the SBis occluded.

A relationship between the size of a coronary vessel and the volume ofmyocardium perfused by the vessel based on a fractal model has beenreported. The infarct index was expressed as the ratio of potentialischemic myocardium and estimated based on the ratio of the radius ofvessel of interest raised to a Murray's exponent to the sum of similarterms for the left and right coronary arteries. Neither Murray'sexponent of 3, nor an exponent of 2.7 are supported by experimental datafor coronary arteries. A value of 7/3 (2.33) has been shown based ondirect experimental measurements based on the minimum energy hypothesis.The disclosure of the present application provides a more basicderivation of this parameter based on actual measurements in the swinecoronary vasculature, resulting in a foundation for the relationshipbetween the SB caliber and perfused myocardial mass.

BRIEF SUMMARY

In at least one embodiment of a method for diagnosing a risk of cardiacdisease, the method comprises the steps of identifying a luminalcross-sectional area of a side branch vessel, identifying a luminalcross-sectional area of a main artery most proximal to the side branchvessel, determining a myocardial region at risk for infarct from sidebranch occlusion relative to a mass perfused by the most proximalartery, wherein such region is based on a relationship between thecross-sectional areas of the side branch and the most proximal mainartery, and diagnosing a risk of cardiac disease based on a diameter ofthe side branch and the size of the myocardial region at risk forinfarct perfused by the side branch. In another embodiment, the step ofidentifying a luminal cross-sectional area of a side branch vessel isperformed by coronary angiography. In yet another embodiment, the mainartery most proximal to the side branch vessel comprises the leftanterior descending artery. In an additional embodiment, the main arterymost proximal to the side branch vessel comprises the left circumflexartery. In a further embodiment, the main artery most proximal to theside branch vessel comprises the right coronary artery.

In at least one embodiment of a method for diagnosing a risk of cardiacdisease according to the present disclosure, the side branch vesselconsists of a diseased side branch vessel. In another embodiment, thestep of identifying a luminal cross-sectional area of a side branchvessel is performed using a diameter of a mother vessel of the sidebranch bifurcation and a diameter of a daughter vessel of the sidebranch bifurcation.

In at least one embodiment of a method for diagnosing a risk of cardiacdisease according to the present disclosure, the method comprises thesteps of identifying a luminal cross-sectional area of a side branchvessel, identifying a luminal cross-sectional area of a left maincoronary artery, identifying a luminal cross-sectional area of a rightcoronary artery, determining a myocardial region at risk for infarctfrom side branch occlusion relative to a mass perfused by the entireheart, wherein such region is based on a relationship between thecross-sectional areas of the side branch, left main coronary artery andright coronary artery, and diagnosing a risk of cardiac disease based ona diameter of the side branch and the size of the myocardial region atrisk for infarct perfused by the side branch. In another embodiment, thestep of identifying a luminal cross-sectional area of a side branchvessel is performed by coronary angiography. In yet another embodiment,the side branch vessel is diseased. In a further embodiment, the step ofidentifying a luminal cross-sectional area of a side branch vessel isperformed using a diameter of a mother vessel of the side branchbifurcation and a diameter of a daughter vessel of the side branchbifurcation.

In at least one embodiment of a system for diagnosing a risk of cardiacdisease, the system comprises a processor, a storage medium operablyconnected to the processor, the storage medium capable of receiving andstoring data relative of measurements from a vasculature of a vessel,wherein the processor is operable to identify a luminal cross-sectionalarea of a side branch vessel, identify a luminal cross-sectional area ofa main artery most proximal to the side branch vessel, determine amyocardial region at risk for infarct from side branch occlusionrelative to a mass perfused by the most proximal artery, wherein suchregion is based on a relationship between the cross-sectional areas ofthe side branch and the most proximal main artery, and diagnose a riskof cardiac disease based on a diameter of the side branch and the sizeof the myocardial region at risk for infarct perfused by the sidebranch. In another embodiment, the identification of a luminalcross-sectional area of a side branch vessel is performed by coronaryangiography. In yet another embodiment, the main artery most proximal tothe side branch vessel comprises the left anterior descending artery. Inan additional embodiment, the main artery most proximal to the sidebranch vessel comprises the left circumflex artery. In a furtherembodiment, the main artery most proximal to the side branch vesselcomprises the right coronary artery.

In at least one embodiment of a system for diagnosing a risk of cardiacdisease according to the present disclosure, the side branch vesselconsists of a diseased side branch vessel. In another embodiment, theidentification of a luminal cross-sectional area of a side branch vesselis performed using a diameter of a mother vessel of the side branchbifurcation and a diameter of a daughter vessel of the side branchbifurcation.

In at least one embodiment of a system for diagnosing a risk of cardiacdisease, the system comprises a processor, a storage medium operablyconnected to the processor, the storage medium capable of receiving andstoring data relative of measurements from a vasculature of a vessel,wherein the processor is operable to identify a luminal cross-sectionalarea of a side branch vessel, identify a luminal cross-sectional area ofa left main coronary artery, identify a luminal cross-sectional area ofa right coronary artery, determine a myocardial region at risk forinfarct from side branch occlusion relative to a mass perfused by theentire heart, wherein such region is based on a relationship between thecross-sectional areas of the side branch, left main coronary artery andright coronary artery, and diagnose a risk of cardiac disease based on adiameter of the side branch and the size of the myocardial region atrisk for infarct perfused by the side branch. In another embodiment, theidentification of a luminal cross-sectional area of a side branch vesselis performed by coronary angiography. In yet another embodiment, theside branch vessel is diseased. In a further embodiment, theidentification of a luminal cross-sectional area of a side branch vesselis performed using a diameter of a mother vessel of the side branchbifurcation and a diameter of a daughter vessel of the side branchbifurcation.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows the relation between normalized cumulative arterial volumeand corresponding normalized cumulative arterial length for each crownon a log-log plot, according to at least one embodiment of the presentdisclosure;

FIG. 2 shows the presence of DCAD at locations along the mean trendlines for normal (solid) and DCAD vasculature (broken) according to atleast one embodiment of the present disclosure;

FIG. 3 shows a diagnostic system according to at least one embodiment ofthe present disclosure;

FIG. 4 shows an illustration of a definition of a stem-crown unitaccording to at least one embodiment of the present disclosure;

FIGS. 5A-5C show relationships between resistance and diameter andnormalized crown length of LAD, LCx, and RCA trees of a pig,respectively, according to at least one embodiment of the presentdisclosure;

FIGS. 5D-5F show relationships between resistance and length of LAD,LCx, and RCA trees of a pig, respectively, according to at least oneembodiment of the present disclosure;

FIG. 6A shows a relationship between resistance and diameter andnormalized crown length in symmetric vascular trees for various species,according to at least one embodiment of the present disclosure;

FIG. 6B shows a relationship between resistance and length in symmetricvascular trees for various species, according to at least one embodimentof the present disclosure;

FIG. 7A shows a table of parameters with correlation coefficientscalculated from the Marquardt-Levenberg algorithm for various species,according to at least one embodiment of the present disclosure;

FIG. 7B shows a comparison of data from nonlinear regression andequations of the present disclosure; according to at least oneembodiment of the present disclosure;

FIG. 8A shows a relationship between resistance and diameter andnormalized crown length in the LAD, LCx, and RCA epicardial trees of apig, respectively, according to at least one embodiment of the presentdisclosure;

FIG. 8B shows a relationship between resistance and length in the LAD,LCx, and RCA epicardial trees of a pig, respectively, according to atleast one embodiment of the present disclosure;

FIG. 9 shows a table of parameters B and A in asymmetric coronary treesand corresponding epicardial trees with vessel diameters greater than 1mm, according to at least one embodiment of the present disclosure;

FIG. 10 shows a table of parameters B and A in various organs, accordingto at least one embodiment of the present disclosure;

FIG. 11 shows a table of parameter A obtained from nonlinear regressionin various organs, according to at least one embodiment of the presentdisclosure;

FIGS. 12A-12C show relations between diameter and length and normalizedcrown volume in the LAD, LCx, and RCA trees of a pig, respectively,according to at least one embodiment of the present disclosure;

FIG. 13 shows a relation between diameter and length and normalizedcrown volume in the LAD, LCx, and RCA epicardial trees of a pig,respectively, according to at least one embodiment of the presentdisclosure;

FIG. 14 shows a relation between diameter and length and normalizedcrown volume in the symmetric vascular tree for various organs andspecies, according to at least one embodiment of the present disclosure;and

FIG. 15 shows a comparison of data from nonlinear regression and anequation of the present disclosure; according to at least one embodimentof the present disclosure;

FIG. 16A shows a relationship between the side branch diameter and thepercentage of myocardial mass at risk for infarct relative to the massperfused by the most proximal artery, according to at least oneembodiment of the present disclosure;

FIG. 16B shows a relationship between the side branch area and thepercentage of myocardial mass at risk for infarct relative to the massperfused by the most proximal artery, according to at least oneembodiment of the present disclosure.

DETAILED DESCRIPTION

For the purposes of promoting an understanding of the principles of thepresent disclosure, reference will now be made to the embodimentsillustrated in the drawings, and specific language will be used todescribe the same. It will nevertheless be understood that no limitationof the scope of this disclosure is thereby intended.

The disclosure of the present application applies concepts frombiomimetics and microfluidics to analyze vascular tree structure, thusimproving the efficacy and accuracy of diagnostics involving vasculardiseases such as DCAD. Scaling laws are developed in the form ofequations that use the relationships between arterial volume,cross-sectional area, blood flow and the distal arterial length toquantify moderate levels of diffuse coronary artery disease. For thepurposes of promoting an understanding of the principles of the presentdisclosure, reference will now be made to the embodiments illustrated inthe drawings, and specific language will be used to describe the same.It will nevertheless be understood that no limitation of the scope ofthe present disclosure is thereby intended.

Biomimetics (also known as bionics, biognosis, biomimicry, or bionicalcreativity engineering) is defined as the application of methods andsystems found in nature to the study and design of engineering systemsand modern technology. The mimic of technology from nature is based onthe premise that evolutionary pressure forces natural systems to becomehighly optimized and efficient. Some examples include (1) thedevelopment of dirt- and water-repellent paint from the observation thatthe surface of the lotus flower plant is practically unsticky, (2) hullsof boats imitating the thick skin of dolphins, and (3) sonar, radar, andmedical ultrasound imaging imitating the echolocation of bats.

Microfluidics is the study of the behavior, control and manipulation ofmicroliter and nanoliter volumes of fluids. It is a multidisciplinaryfield comprising physics, chemistry, engineering and biotechnology, withpractical applications to the design of systems in which such smallvolumes of fluids may be used. Microfluidics is used in the developmentof DNA chips, micro-propulsion, micro-thermal technologies, andlab-on-a-chip technology.

Regarding the minimum energy hypothesis, the architecture (or manifolds)of the transport network is essential for transport of material inmicrofluid channels for various chips. The issue is how to design newdevices, and more particularly, how to fabricate microfluidic channelsthat provide a minimum cost of operation. Nature has developed optimalchannels (or transport systems) that utilize minimum energy fortransport of fluids. The utility of nature's design of transport systemsin engineering applications is an important area of biomimetics.

Biological trees (for example, vascular trees) are either used toconduct fluids such as blood, air, bile or urine. Energy expenditure isrequired for the conduction of fluid through a tree structure because offrictional losses. The frictional losses are reduced when the vesselbranches have larger diameters. However, this comes with a costassociated with the metabolic construction and maintenance of the largervolume of the structure. The question is what physical or physiologicalfactors dictate the design of vascular trees. The answer is that thedesign of vascular trees obeys the “minimum energy hypothesis”, i.e.,the cost of construction and operation of the vascular system appears tobe optimized.

The disclosure of the present application is based on a set of scalinglaws determined from a developed minimum energy hypothesis. Equation #1(the “volume-length relation”) demonstrates a relationship betweenvessel volume, the volume of the entire crown, vessel length, and thecumulative vessel length of the crown:

$\begin{matrix}{\frac{V}{V_{m\; {ax}}} = \left( \frac{L}{L_{m\; {ax}}} \right)^{\frac{5}{ɛ^{\prime} + 1}}} & (1)\end{matrix}$

In Equation #1, V represents the vessel volume, V_(max) the volume ofthe entire crown, L represents the vessel length, L_(max) represents thecumulative vessel length of the entire crown, and ε′ represents thecrown flow resistance, which is equal to the ratio of metabolic toviscous power dissipation.

Equation #2 (the “diameter-length relation”) demonstrates a relationshipbetween vessel diameter, the diameter of the most proximal stem, vessellength, and the cumulative vessel length of the crown:

$\begin{matrix}{\frac{D}{D_{m\; {ax}}} = \left( \frac{L}{L_{\; {m\; {ax}}}} \right)^{\frac{{3ɛ^{\prime}} - 2}{4{({ɛ^{\prime} + 1})}}}} & (2)\end{matrix}$

In Equation #2, D represents the vessel diameter, D_(max) represents thediameter of the most proximal stem, L represents the vessel length,L_(max) represents the cumulative vessel length of the entire crown, andε′ represents the crown flow resistance, which is equal to the ratio ofmetabolic to viscous power dissipation.

Equation #3 (the “flow rate-diameter relation”) demonstrates arelationship between the flow rate of a stem, the flow rate of the mostproximal stem, vessel diameter, and the diameter of the most proximalstem:

$\begin{matrix}{\frac{Q}{Q_{m\; {ax}}} = \left( \frac{D}{D_{m\; {ax}}} \right)^{\frac{4{({ɛ^{\prime} + 1})}}{{3ɛ^{\prime}} - 2}}} & (3)\end{matrix}$

In Equation #3, Q represents flow rate of a stem, Q_(max) represents theflow rate of the most proximal stem, V represents vessel diameter,V_(max) represents the diameter of the most proximal stem, and ε′represents the crown flow resistance, which is equal to the ratio ofmetabolic to viscous power dissipation.

Regarding the aforementioned Equations, a vessel segment is referred toas a “stem,” and the entire tree distal to the stem is referred as a“crown.” The aforementioned parameters relate to the crown flowresistance and is equal to the ratio of maximum metabolic-to-viscouspower dissipation.

Two additional relations were found for the vascular trees. Equation #4(the “resistance-length and volume relation”) demonstrates arelationship between the crown resistance, the resistance of the entiretree, vessel length, the cumulative vessel length of the crown, vesselvolume, and the volume of the entire crown:

$\begin{matrix}{\frac{R_{c}}{R_{{ma}\; x}} = \frac{\left( {L/L_{m\; {ax}}} \right)^{3}}{\left( {V/V_{m\; {ax}}} \right)^{ɛ^{\prime\prime}}}} & (4)\end{matrix}$

In Equation #4, R_(c) represents the crown resistance, R_(max)represents the resistance of the entire tree, L represents vessellength, L_(max) represents the cumulative vessel length of the entirecrown, V represents vessel volume, V_(max) represents the volume of theentire crown, and ε′ represents the crown flow resistance, which isequal to the ratio of metabolic to viscous power dissipation.Resistance, as referenced herein, is defined as the ratio of pressuredifferenced between inlet and outlet of the vessel.

Equation #5 (the “flow rate-length relation”) demonstrates arelationship between the flow rate of a stem, the flow rate of the mostproximal stem, vessel length, the cumulative vessel length of the entirecrown:

$\begin{matrix}{\frac{Q}{Q_{m\; {ax}}} = \frac{L}{L_{m\; {ax}}}} & (5)\end{matrix}$

In Equation #5, Q represents flow rate of a stem, Q_(max) represents theflow rate of the most proximal stem, L represents vessel length, andL_(max) represents the cumulative vessel length of the entire crown.

In at least one embodiment of the disclosure of the present application,the application of one or more of the aforementioned Equations toacquired vessel data may be useful diagnose and/or aid in the diagnosisof disease.

By way of example, the application of one or more of the aforementionedEquations are useful to diagnose DCAD. For such a diagnosis, theapplications of Equations #1-#3 may provide the “signatures” of normalvascular trees and impart a rationale for diagnosis of diseaseprocesses. The self-similar nature of these laws implies that theanalysis can be carried out on a partial tree as obtained from anangiogram, a computed tomography (CT) scan, or an magnetic resonanceimaging (MRI). Hence, the application of these Equations to the obtainedimages may serve for diagnosis of vascular disease that affect the lumendimension, volume, length (vascularity) or perfusion (flow rate).Additionally, the fabrication of the microfluidic channels can begoverned by Equations #1-#5 to yield a system that requires minimumenergy of construction and operation. Hence, energy requirements will beat a minimum to transport the required microfluidics.

In one exemplary embodiment, the application of the volume-lengthrelation (Equation #1) to actual obtained images is considered as shownin FIG. 1. First, images (angiograms in this example) of swine coronaryarties were obtained. The application of Equation #1 on various volumesand lengths from the angiograms resulted in the individual data pointsshown within FIG. 1 (on a logarithmic scale). The line depicted withinFIG. 1 represents the mean of the data points (the best fit) among theidentified data points.

In FIG. 2, the mean of the data (solid line) is compared to an animalwith diffuse disease at three different vessel sizes: proximal (1),middle (2), and distal (3). The reductions in volume shown on FIG. 2correspond to approximately 40% stenosis, which is typicallyundetectable with current methodologies. At each diffuse stenosis, thelength remains constant but the diameter (cross-sectional, and hence,volume) changes. The length is unlikely to change unless the flowbecomes limiting (more than approximately 80% stenosis) and the vascularsystem experiences vessel loss (rarefication) and remodeling. It isclear that a 40% stenosis deviates significantly from the y-axis (asdetermined by statistical tests) from the normal vasculature, and assuch, 40% stenosis can be diagnosed by the system and method of thedisclosure of the present application. It can be appreciated that thedisclosure of the present application can predict inefficiencies as lowas about 10%, compared to well-trained clinicians who can only predictinefficiencies at about 60% at best.

This exemplary statistical test compares the deviation of disease tonormality relative to the variation within normality. The location ofthe deviation along the x-axis corresponds to the size of the vessel.The vessel dimensions range as proximal>mid>distal. Hence, by utilizingthe system and method of the disclosure of the present application, thediagnosis of the extent of disease and the dimension of the vesselbranch is now possible. Similar embodiments with other scaling relationsas described herein can be applied similarly to model and actualvascular data.

The techniques disclosed herein have tremendous application in a largenumber of technologies. For example, a software program or hardwaredevice may be developed to diagnose the percentage of inefficiency(hence, occlusion) in a circulatory vessel or system.

Regarding the computer-assisted determination of such diagnoses, anexemplary system of the disclosure of the present application isprovided. Referring now to FIG. 3, there is shown a diagrammatic view ofan embodiment of diagnostic system 300 of the present disclosure. In theembodiment shown in FIG. 3, diagnostic system 300 comprises user system302. In this exemplary embodiment, user system 302 comprises processor304 and one or more storage media 306. Processor 304 operates upon dataobtained by or contained within user system 302. Storage medium 306 maycontain database 308, whereby database 308 is capable of storing andretrieving data. Storage media 306 may contain a program (including, butnot limited to, database 308), the program operable by processor 304 toperform a series of steps regarding data relative of vessel measurementsas described in further detail herein.

Any number of storage media 306 may be used with diagnostic system 300of the present disclosure, including, but not limited to, one or more ofrandom access memory, read only memory, EPROMs, hard disk drives, floppydisk drives, optical disk drives, cartridge media, and smart cards, forexample. As related to user system 302, storage media 306 may operate bystoring data relative of vessel measurements for access by a user and/orfor storing computer instructions. Processor 304 may also operate upondata stored within database 308.

Regardless of the embodiment of diagnostic system 300 referenced hereinand/or contemplated to be within the scope of the present disclosure,each user system 302 may be of various configurations well known in theart. By way of example, user system 302, as shown in FIG. 3, compriseskeyboard 310, monitor 312, and printer 314. Processor 304 may furtheroperate to manage input and output from keyboard 310, monitor 312, andprinter 314. Keyboard 310 is an exemplary input device, operating as ameans for a user to input information to user system 302. Monitor 312operates as a visual display means to display the data relative ofvessel measurements and related information to a user using a usersystem 302. Printer 314 operates as a means to display data relative ofvessel measurements and related information. Other input and outputdevices, such as a keypad, a computer mouse, a fingerprint reader, apointing device, a microphone, and one or more loudspeakers arecontemplated to be within the scope of the present disclosure. It can beappreciated that processor 304, keyboard 310, monitor 312, printer 314and other input and output devices referenced herein may be componentsof one or more user systems 302 of the present disclosure.

It can be appreciated that diagnostic system 300 may further compriseone or more server systems 316 in bidirectional communication with usersystem 302, either by direct communication (shown by the single lineconnection on FIG. 3), or through a network 318 (shown by the doubleline connections on FIG. 3) by one of several configurations known inthe art. Such server systems 316 may comprise one or more of thefeatures of a user system 302 as described herein, including, but notlimited to, processor 304, storage media 306, database 308, keyboard310, monitor 312, and printer 314, as shown in the embodiment ofdiagnostic system 300 shown in FIG. 3. Such server systems 316 may allowbidirectional communication with one or more user systems 302 to allowuser system 302 to access data relative of vessel measurements andrelated information from the server systems 316. It can be appreciatedthat a user system 302 and/or a server system 316 referenced herein maybe generally referred to as a “computer.”

Several concepts are defined to formulate resistance scaling laws of thedisclosure of the present application. A vessel segment is defined as a“stem” and the entire tree distal to the stem is defined as a “crown,”as shown in FIG. 4 and as previously disclosed herein. FIG. 4 shows aschematic illustration of the definition of the stem-crown unit. Threestem-crown units are shown successively (1, 2, and n), with the smallestunit corresponding to an arteriole-capillary or venule-capillary unit.An entire vascular tree, or substantially the entire vascular tree,consists of many stem-crown units down to, for example, the smallestarterioles or venules. In one exemplary embodiment of the disclosure ofthe present application, the capillary network (referenced herein ashaving vessel diameters of less than 8 microns) is excluded from theanalysis because it is not tree-like in structure. A stem, for purposesof simplification, is assumed to be a cylindrical tube with noconsideration of vessel tapering and other nonlinear effects as theyplay a relatively minor role in determining the hemodynamics of theentire tree. However, the disclosure of the present application is notintended to be limited by the aforementioned capillary network exclusionand/or the aforementioned stem assumption.

Through the Hagen-Poiseuille law known in the art, the resistance of thesteady laminar flow in a stem of an entire tree may be provided as shownin Equation #6:

$\begin{matrix}{R_{s} = \frac{\Delta \; P_{s\;}}{Q_{s}}} & (6)\end{matrix}$

In Equation #6, R_(s) is the resistance of a stem segment, ΔP_(s) is thepressure gradient along the stem, and Q_(s) is a volumetric flow ratethrough the stem.

According to the disclosure of the present application, Equation #6,providing for R_(s) may be written in a form considering stem length anddiameter, as shown in Equation #7.

$\begin{matrix}{R_{s} = {\frac{128\; \mu \; L_{s}}{\pi \; D_{s}^{4}} = {K_{s}\frac{L_{s}}{D_{s}^{4}}}}} & (7)\end{matrix}$

In Equation #7, R_(s) is the resistance of a stem segment, L_(s) is thelength of the stem, D_(s) is the diameter of the stem, μ is theviscosity of a fluid, and K_(s) is a constant equivalent to 128μ/π.

Furthermore, the resistance of a crown may be demonstrated as shown inEquation #8:

$\begin{matrix}{R_{c} = \frac{\Delta \; P_{c}}{Q_{s}}} & (8)\end{matrix}$

In Equation #8, R_(c) is the crown resistance, ΔP_(s) is the pressuregradient in the crown from the stem to the terminal vessels, and Q_(s)is a volumetric flow rate through the stem. Equation #8 may also bewritten in a novel form to solve for R_(c) in accordance with thedisclosure of the present application as shown in Equation #9:

$\begin{matrix}{R_{c} = {K_{c}\frac{L_{c}}{D_{s}^{4}}}} & (9)\end{matrix}$

In Equation #9, R_(c) is the crown resistance, L_(c) is the crownlength, D_(s) is the diameter of the stem vessel proximal to the crown,and K_(c) is a constant that depends on the branching ration, diameterratio, the total number of tree generations, and viscosity in the crown.The crown length, L_(c), may be defined as the sum of the lengths ofeach vessel in the crown (or substantially all of the vessels in thecrown).

As Equation #9, according to the disclosure of the present application,is applicable to any stem-crown unit, one may obtain the followingequation:

$\begin{matrix}{R_{{ma}\; x} = {K_{c}\frac{L_{m\; {ax}}}{D_{m\; {ax}}^{4}}}} & (10)\end{matrix}$

so that the following formula for K_(c) may be obtained:

$\begin{matrix}{K_{c} = \frac{R_{m\; {ax}} \cdot D_{m\; {ax}}^{4}}{L_{m\; {ax}}}} & (11)\end{matrix}$

D_(max), L_(max), and R_(max) correspond to the most proximal stemdiameter, the cumulative vascular length, and total resistance of theentire tree, respectively. In the non-dimensional form, Equation #11 canbe written as:

$\begin{matrix}{{\left( \frac{R_{c}}{R_{m\; {ax}}} \right) \cdot \left( \frac{D_{s}}{D_{m\; {ax}}} \right)^{4}} = {A_{1}\left( \frac{L_{c}}{L_{m\; {ax}}} \right)}} & (12)\end{matrix}$

Parameter A₁ in Equation #12, as provided above, should be equal to one.From Equations #7 and #9, one may then obtain the desired resistancescaling relation between a single vessel (a stem) and the distal crowntree:

$\begin{matrix}{\left( \frac{R_{s}}{R_{c}} \right) = {\frac{K_{s}}{K_{c}}\left( \frac{L_{s}}{L_{c}} \right)}} & (13)\end{matrix}$

Equations #7-13 relate the resistance of a single vessel to thecorresponding distal tree,

Verification. The asymmetric coronary arterial trees of hearts andsymmetric vascular trees of many organs were used to verify the proposedresistance scaling law. First, the asymmetric coronary arterial tree hasbeen reconstructed in pig hearts by using the growth algorithmintroduced by Mittal et al. (A computer reconstruction of the entirecoronary arterial tree based on detailed morphometric data. Ann. Biomed.Eng. 33 (8):1015-1026 (2005)) based on measured morphometric data ofKassab et al. (Morphometry of pig coronary arterial trees. Am J PhysiolHeart Circ Physiol. 265:H350-H365 (1993)). Briefly, vessels ≧40 μm werereconstructed from cast data while vessels <40 μm were reconstructedfrom histological data. After the tree was reconstructed, each vesselwas assigned by diameter-defined Strahler orders which was developedbased on the Strahler system (Strahler, A. N. Hypsometric (areaaltitude) analysis of erosional topology. Bull Geol Soc Am. 63:1117-1142(1952)).

Furthermore, symmetric vascular trees of many organs were constructed inthe Strahler system, based on the available literature. Here, thepulmonary arterial tree of rats was obtained from the study of Jiang etal. (Diameter-defined Strahler system and connectivity matrix of thepulmonary arterial tree. J. Appl. Physiol. 76:882-892 (1994)); thepulmonary arterial/venous trees of cats from Yen et al. (Morphometry ofcat's pulmonary arterial tree. J. Biomech. Eng. 106:131-136 (1984) andMorphometry of cat pulmonary venous tree. J. Appl. Physiol. Respir.Environ. Exercise. Physiol. 55:236-242 (1983)); the pulmonary arterialtrees of humans from Singhal et al. (Morphometric study of pulmonaryarterial tree and its hemodynamics. J. Assoc. Physicians India,21:719-722 (1973) and Morphometry of the human pulmonary arterial tree.Circ. Res. 33:190 (1973)) and Huang et al. (Morphometry of the humanpulmonary vasculature. J. Appl. Physiol. 81:2123-2133 (1996)); thepulmonary venous trees of humans from Horsfield et al. (Morphometry ofpulmonary veins in man. Lung. 159:211-218 (1981)) and Huang et al.; theskin muscle arterial tree of hamsters from Bertuglia et al, (Hypoxia- orhyperoxia-induced changes in arteriolar vasomotion in skeletal musclemicrocirculation. Am J Physiol Heart Circ Physiol. 260:H362-H372(1991)); the retractor muscle arterial tree of hamsters from Ellsworthet al. (Analysis of vascular pattern and dimensions in arteriolarnetworks of the retractor muscle in young hamsters. Microvasc. Res.34:168-183 (1987)); the mesentery arterial tree of rats from Ley et al.(Topological structure of rat mesenteric microvessel networks.Microvasc. Res. 32:315-332 (1986)); the sartorius muscle arterial treeof cats from Koller et al. (Quantitative analysis of arteriolar networkarchitecture in cat sartorius muscle. Am J Physiol Heart Circ Physiol.253: H154-H164 (1987)); and the bulbular conjunctiva arterial/venoustrees of humans and the omentum arterial tree of rabbits from Fenton etal. (Microcirculatory model relating geometrical variation to changes inpressure and flow rate. Ann. Biomed. Eng. 1981; 9:303-321 (1981)).

Data analysis. For the asymmetric coronary arterial trees, full treedata are presented as log-log density plots showing the frequency ofdata because of the enormity of data points, i.e., darkest shadereflects highest frequency or density and the lightest shade reflectsthe lowest frequency. The nonlinear regression (SigmaStat 3.5) is usedto analyze the data in both asymmetric and symmetric tree, which usesthe Marquardt-Levenberg algorithm (nonlinear regression) to find thecoefficients (parameters) of the independent variables that give the“best fit” between the equation and the data.

Results: Validation of resistance scaling law in entire vascular trees.The predictions of these novel scaling laws were then validated in boththe asymmetric coronary trees and the symmetric vascular trees for whichthere exists morphometric data in the literature (e.g., vessels ofvarious skeletal muscles, mesentery, omentum, and conjunctiva).

First, the entire asymmetric coronary LAD, LCx, and RCA trees withseveral millions of vessels were analyzed(15,16). FIGS. 5A, 5B, and 5Cshow a log-log plot of (R_(c)/R_(max))·(D_(s)/D_(max))⁴ as a function ofnormalized crown length (L_(c)/L_(max)) for LAD, LCx, and RCA trees,respectively. Relationships between (R_(c)/R_(max))·(D_(s)/D_(max))⁴ andnormalized crown length (L_(c)/L_(max)) in the asymmetric entire LAD(FIG. 5A), LCx (FIG. 5B), and RCA (FIG. 5C) trees of pig, which include946937, 571383, and 836712 stem-crown units are shown, respectively.Through the Marquardt-Levenberg algorithm with the exponents ofL_(c)/L_(max) constrained to one, parameter A₁ in Equation #12 has avalue of 1.027 (R²=0.990), 0.993 (R²=0.997), and 1.084 (R²=0.975) forLAD, LCx, and RCA trees, respectively. The values of A₁ obtained frommorphometric data are in agreement with the theoretical value of one.Corresponding to FIGS. 5A, 5B, and 5C, FIGS. 5D, 5E, and 5F show alog-log plot of R_(c)/R_(s) as a function of L_(c)/L_(s). ParameterK_(s)/K_(c) in Equation #13 has a value of 2.647 (R²=0.954), 2.943(R²=0.918), and 2.147 (R²=0.909) for LAD, LCx, and RCA trees,respectively. FIGS. 5D, 5E, and 5F show a relationship betweenR_(c)/R_(s) and L_(c)/L_(s) in the LAD, LCx, and RCA trees of pig,corresponding to FIGS. 5A, 5B, and 5C.

Furthermore, FIGS. 6A and 6B show the log-log plots of(R_(c)/R_(max))·(D₀/D_(max))⁴ and R_(c)/R_(s) as a function ofL_(c)/L_(max) and L_(c)/L_(s), respectively, in the vascular trees ofvarious species. Corresponding to FIGS. 6A and 6B, theMarquardt-Levenberg algorithm was used to calculate the parameters A₁and K_(s)/K_(c) in Equations #12 and #13, respectively, while theexponents of L_(c)/L_(max) and L_(c)/L_(s) were constrained to be one.Parameters A₁ in Equation #12 and K_(s)/K_(c) in Equation #13 withcorrelation coefficient for various species are listed in the tableshown in FIG. 7A. The data in FIG. 7A have a mean value (averaged overall organs and species) of 1.01±0.06 for parameter A₁. FIG. 7B shows acomparison of (K_(s)/K_(c))_(ML) from the nonlinear regression ofanatomical data and (K_(s)/K_(c))_(EQ) based on Equations K_(s)=128μ/πand

${K_{c} = \frac{R_{m\; {ax}} \cdot D_{m\; {ax}}^{4}}{L_{m\; {ax}}}},$

noting that the comparison can be represented as

$\left( \frac{K_{s}}{K_{c}} \right)_{EQ} = {A \cdot {\left( \frac{K_{s}}{K_{c}} \right)_{ML}^{B}.}}$

When A is constrained to be one in the Marquardt-Levenberg algorithm, Bhas a value of one (R²=0.983). Using the same Marquardt-Levenbergalgorithm, a nonlinear regression fit of all raw data yields a mean of1.01 (R²=0.95) for parameter A₁. Both the mean value and the nonlinearregression fit of all data agree with the theoretical value of one.

FIG. 6B shows much smaller R_(c)/R_(s) in pulmonary vascular tree thanother organs at the same value of L_(c)/L_(s). Accordingly, theK_(s)/K_(c) values (shown in the table in FIG. 7A) are similar exceptfor the pulmonary vasculature with a larger value. The K_(s)/K_(c)values are also calculated based on Equations K_(s)=128μ/π andK_(c)=R_(max)·D_(max) ⁴/L_(max), which is compared with the K_(s)/K_(c)values obtained from the Marquardt-Levenberg algorithm, as shown in FIG.7B. The viscosity is determined based on an empirical in vivo relationthat depends on the vessel diameter. The comparison shows goodagreement. The K_(s)/K_(c) values in the pulmonary vasculature have alarger value because the cross-section area of pulmonary tree has alarge increase from proximal to terminal vessels in the pulmonary treeand the resistance of the entire tree (R_(max)) is much smaller. Theagreement between experimental measurement and theoretical relationsillustrate that the novel resistance scaling law disclosed herein ofEquations #9, #12, and #13 can be applied to a general vascular treedown to the smallest arterioles or venules.

Results: Resistance scaling law of partial vascular trees. FIGS. 8A and8B show the relations between (R_(c)/R_(max))·(D_(s)/D_(max))⁴ andnormalized crown volume (L_(c)/L_(max)) and between R_(c)/R_(s) andL_(c)/L_(s), respectively, in the LAD, LCx, and RCA epicardial trees.FIG. 8A shows a relationship between (R_(c)/R_(max))·(D_(s)/D_(max))⁴and normalized crown volume (L_(c)/L_(max)) in the LAD, LCx, and RCAepicardial trees of pig with diameter of mother vessels larger than 1mm, which include 132, 90, and 192 vessel segments, respectively. FIG.8B shows a relationship between R_(c)/R_(s) and L_(c)/L_(s) in the LAD,LCx, and RCA epicardial trees of pig corresponding to FIG. 8A. ParameterA₁ in Equation #12 has a value of 0.902 (R²=0.907), 0.895 (R²=0.887),and 1.000 (R²=0.888) and parameter K_(s)/K_(c) in Equation #13 has avalue of 3.29 (R²=0.875), 3.48 (R²=0.816), and 3.12 (R²=0.927) for theLAD, LCx, and RCA epicardial trees, respectively.

The aforementioned study validates the novel resistance scaling law ofthe present disclosure that relates the resistance of a vessel branch tothe equivalent resistance of the corresponding distal tree in variousvascular trees of different organs and species. The significance of theresistant scaling law is that the hydraulic resistance of a distalvascular tree can be estimated from the proximal vessel segment. As aresult, the disclosure of the present application has wide implicationsfrom understanding fundamental vascular design to diagnosis of diseasein the vascular system.

Resistance scaling law. The mechanisms responsible for blood flowregulation in vascular trees are of central importance, but are stillpoorly understood. The arteriolar beds are the major site of vascularresistance, which contributes to the maintenance and regulation ofregional blood flow. Although arteriolar resistance plays an importantrole in the etiology of many diseases, in particular, hypertension, ithas been difficult to predict the resistance in the arteriolar beds. Thenovel resistance scaling law of the present disclosure addresses thisissue.

The resistance scaling laws (Equations #9, #12, and #13) are derivedbased on the relation of diameter ratio (DR=D_(i)/D_(i-1)), length ratio(LR=L_(i)/L_(i-1)) and branching ratio (BR=N_(i)/N_(i-1)) in a symmetrictree as:

${{DR} = {{{BR}^{\frac{1}{2 + ɛ}}\mspace{14mu} {and}\mspace{14mu} {LR}} = {BR}^{\frac{1}{3}}}},$

where ε=0 and ε=1 represent the area-preservation, πD_(i-1) ²=BR·πD_(i)², and Murray's law, πD_(i-1) ³=BR·πD_(i) ³, respectively.

Although the total cross-sectional area (CSA) may increase dramaticallyfrom the aorta to the arterioles, the variation is significantly smallerin most organs except for the lung. The increase of CSA towards thecapillaries is typically inferred from the decrease in velocity. Thevelocity between the most proximal and distal levels in various organsof mammals is found to vary by about a factor of five, except for thepulmonary vascular trees. This is clearly reflected by the table shownin FIG. 7A, in which

${K_{s}/K_{c}} = \frac{1}{K_{ɛ}}$

is relatively small except for the pulmonary vasculature. This impliesthat wall shear stress (WSS) increases from the arteries to thearterioles in most organs, which is consistent with previousmeasurements.

Structure-function scaling laws obtained from resistance scaling law. Amathematical model (the ¾-power scaling law) was derived in a symmetricvasculature to characterize the allometric scaling laws, based on theminimum energy theory. The ¾-power scaling law can be written asQ_(s)∝M^(3/4), where Q_(s) is the volumetric flow rate of the aorta andM is body mass. In a stem-crown unit, Q_(s) is the volumetric flow rateof the stem and M is the mass perfused by the stem crown unit. Thevolumetric flow rate of the stem is Q_(s)=πD_(s) ²U_(s)/4, where D_(s)and U_(s) are the diameter and the mean flow velocity of the stem(averaged over the cross-section of stem). Similar to at least one knownmodel, the pressure drop from the stem to the capillaries (ΔP_(c)) andthe mean flow velocity of the stem (U_(s)) are independent of theperfused mass so that D_(s)∝M^(3/8) and the resistance of the crown(R_(c)=ΔP_(c)/Q_(s)) is inversely proportional to the volumetric flowrate (R_(c)∝Q_(s) ⁻¹∝M^(−3/4)). Since D_(s)∝M^(3/8), R_(c)∝M^(−3/4),K_(c) is a constant, Equations #9 and #12 yields that the crown lengthL_(c)∝M^(3/4). The cumulative length-mass scaling in pig hearts,L_(c)∝M^(3/4), has recently been verified by the present inventors andtheir research group. This relation, in conjunction with the flow-massrelation (Q_(s)∝M^(3/4)), yields the flow-length relation (Q_(s)∝L_(c))in the stem-crown unit, which has been previously validated.

Here, the crown length L_(c)∝M^(3/4) is different from the biologicallength l∝M^(1/4). The biological length (l) is the cumulative lengthalong a path from inlet (level zero) to the terminal (level N), but thecrown length is the total length of all vessels from inlet to theterminals. Although the biological length shows that the vascularphysiology and anatomy are four-dimensional, the crown length depicts a¾-power relation between the total length of entire/partial biologicalsystem and the perfused mass.

Clinical implications of resistance scaling law: The self-similar natureof the structure-function scaling laws in Equations #9, #12 and #13implies that they can be applied to a partial tree clinically (e.g., apartial tree obtained from an angiogram, computerized tomography, ormagnetic resonance imaging). As provided herein, the hypothesis usingthe LAD, LCx, and RCA epicardial pig trees obtained from casts truncatedat 1 mm diameter to mimic the resolution of noninvasive imagingtechniques was verified. The good agreement between experiments andtheory, as shown in FIG. 8, illustrates that the resistance scaling lawscan be applied to partial vascular trees as well as entire trees.

Significance of resistance scaling law: The novel resistance scaling law(Equations #9 and #12) provides a theoretical and physical basis forunderstanding the hemodynamic resistance of the entire tree (or asubtree) as well as to provide a rational for clinical diagnosis. Thescaling law illustrates the relationship between the structure (tree)and function (resistance), in which the crown resistance is proportionalto the crown length and inversely proportional to the fourth power ofstem diameter D_(s) ⁴. The small crown resistance corresponds to a smallcrown length, thus matching the transport efficiency of the crown. Anincrease of stem diameter can decrease the resistance, which maycontribute to the self scaling of biological transport system. The novelscaling law provides an integration between a single unit and the whole(millions of units) and imparts a rationale for diagnosis of diseaseprocesses as well as assessment of therapeutic trials.

The disclosure of the present application provides a novel volumescaling law in a vessel segment and its corresponding distal tree ofnormal organs and in various species as, for example, V_(c)=K_(v)D_(s)^(2/3)L_(c), where V_(c) and L_(c) are the vascular volume and length,respectively, D_(s) is the diameter of vessel segment, and K_(v) is aconstant. A novel scaling relation of the disclosure of the presentapplication is validated with available vascular morphometric tree data,and may serve as a control reference to examine the change of bloodvolume in various organs under different states using conventionalimaging. A novel scaling law of the disclosure of the presentapplication is further validated through diameter-length, volume-length,flow-diameter, and volume-diameter scaling relations, derived based on aminimum energy hypothesis (15). Hence, the novel volume scaling law ofthe disclosure of the present application is consistent with a (minimumenergy) state of efficient vascular system.

In addition to the foregoing, it is known that V_(c)∝M (M is the massperfused by the stem-crown unit) from the ¾ allometric scaling law,where V_(c) is the crown volume (i.e., the sum of all vessel volumes inthe crown). Therefore, V_(c) can be represented as follows:

V _(c) =C _(v) M ^(1/4) M ^(3/4)  (14)

where C_(v) is a volume-mass constant.

There are two scaling relations: stem diameter-mass relation,D_(s)∝M^(3/8), wherein D_(s) is the diameter of stem vessel, and crownlength-mass relation, L_(c)∝M^(3/4), wherein L_(c) is the crown lengththat is defined as the sum of the lengths or substantially all of thelengths of each vessel in the crown).

From D_(s)=C_(d)M^(3/8), L_(c)=C_(l)M^(3/4), and Equation #14, one mayobtain:

$\begin{matrix}{{V_{c} = {{C_{v}M^{1/4}M^{3/4}} = {C_{v}\left( \frac{D_{s}}{C_{d}} \right)}^{2/3}}}{\frac{L_{c}}{C_{l}} = {K_{v}D_{s}^{2/3}L_{c}}}} & (15)\end{matrix}$

where K_(v)=C_(v)/(C_(d) ^(2/3)C_(l)) is a constant. Since Equation #15is applicable to any stem-crown unit, one may obtainV_(max)=K_(v)D_(max) ^(2/3)L_(max), so that

${K_{v} = \frac{V_{m\; {ax}}}{D_{m\; {ax}}^{2/3}L_{m\; {ax}}}},$

where D_(max), L_(max), and V_(max) correspond to the most proximal stemdiameter, the cumulative vascular length of entire tree, and thecumulative vascular volume of entire tree, respectively. Equation #15can also be made non-dimensional as:

$\begin{matrix}{\left( \frac{V_{c}}{V_{m\; {ax}}} \right) = {\left( \frac{D_{s}}{D_{m\; {ax}}} \right)^{\frac{2}{3}}\left( \frac{L_{c}}{L_{m\; {ax}}} \right)}} & (16)\end{matrix}$

Morphometry of Vascular Trees. The volume scaling law of the disclosureof the present application is validated in the asymmetric entirecoronary arterial tree reconstructed in pig hearts through the growthalgorithm based on measured morphometric data. Furthermore, theasymmetric epicardial coronary arterial trees with vessel diametergreater than 1 mm were used to validate the scaling laws in partialvascular trees to mimic the resolution of medical imaging.

Symmetric vascular trees of many organs down to the smallest arterioleswere used to verify the proposed structure-function scaling law, whichwere constructed in the Strahler system, based on the availableliterature. The arterial and/or venous trees from the various specieswere obtained as previously referenced herein.

Data Analysis. All scaling relations (i.e., Equations #16 and #29-32)can be represented by a form of the type:

Y=A·X ^(B)  (17)

where X and Y are defined such that A and B should have theoreticalvalues of unity for Equation #16. X and Y are defined as

${\left( \frac{D_{s}}{D_{m\; {ax}}} \right)^{\frac{2}{3}}\left( \frac{L_{c}}{L_{m\; {ax}}} \right)\mspace{14mu} {and}\mspace{14mu} \left( \frac{V_{c}}{V_{m\; {ax}}} \right)},$

respectively. For Equations #29-32, X and Y are defined as

${\left( \frac{L_{c}}{L_{m\; {ax}}} \right)\mspace{14mu} {and}\mspace{14mu} \left( \frac{D_{s}}{D_{m\; {ax}}} \right)};{\left( \frac{L_{c}}{L_{m\; {ax}}} \right)\mspace{14mu} {and}\mspace{14mu} \left( \frac{V_{c}}{V_{m\; {ax}}} \right)};{{and}\mspace{14mu} \left( \frac{D_{s}}{D_{m\; {ax}}} \right)\mspace{14mu} {{and}\left( \frac{Q_{s}}{Q_{m\; {ax}}} \right)}};{\left( \frac{D_{s}}{D_{m\; {ax}}} \right)\mspace{14mu} {and}\mspace{14mu} \left( \frac{V_{c}}{V_{m\; {ax}}} \right)};$

respectively.

A nonlinear regression was then used to calculate A with B constrainedto 3/7, 1 2/7, 2⅓, and 3 for Equations #29-32, respectively. Thenonlinear regression uses the Marquardt-Levenberg algorithm to find theparameter, A, for the variables X and Y to provide the “best fit”between the equation and the data. In Equations #16 and #29-32, theparameter A should have a theoretical value of one.

Results.

Asymmetric Tree Model. The disclosure of the present applicationprovides a novel volume scaling law that relates the crown volume to thestem diameter and crown length in Equations #15 and #16. The validity ofEquations #15 and #16 were examined in the asymmetric entire (down tothe pre-capillary vessel segments) and epicardial (vessel diameter ≧1mm) LAD, LCx, and RCA trees of pig, as shown in FIGS. 12 and 13,respectively. FIG. 12 shows a relation between

$\left( \frac{D_{s}}{D_{\; {m\; {ax}}}} \right)^{\frac{2}{3}}\left( \frac{L_{c}}{L_{m\; {ax}}} \right)$

and normalized crown volume in the entire asymmetric (a) LAD, (b) LCx,and (c) RCA trees of pig, which include 946,937, 571,383, and 836,712vessel segments, respectively. The entire tree data are presented aslog-log density plots showing the frequency of data because of theenormity of data points, i.e., darkest shade reflects highest frequencyor density and the lightest shade reflects the lowest frequency. FIG. 13shows a relation between

$\left( \frac{D_{s}}{D_{\; {m\; {ax}}}} \right)^{\frac{2}{3}}\left( \frac{L_{c}}{L_{m\; {ax}}} \right)$

and normalized crown volume in the asymmetric LAD, LCx, and RCAepicardial trees of pig with vessel diameter larger than 1 mm, whichinclude 66, 42, and 71 vessel segments, respectively.

As shown in FIG. 9, exponent B is determined from a least-square fit,and parameter A is calculated by the nonlinear regression with theexponent B constrained to one. Both B and A for the entire asymmetricand partial trees show agreement with the theoretical value of one. Forthe table shown in FIG. 9, Parameters B (obtained from least-squarefits) and A (obtained from nonlinear regression with B constrained toone) in the asymmetric entire coronary trees and in the correspondingepicardial trees with vessel diameter >1 mm when Equation #16 isrepresented by Equation #17, where independent variables

${X = {{\left( \frac{D_{s}}{D_{m\; {ax}}} \right)^{\frac{2}{3}}\left( \frac{L_{c}}{L_{m\; {ax}}} \right)\mspace{14mu} {and}\mspace{14mu} Y} = \left( \frac{V_{c}}{V_{m\; {ax}}} \right)}},$

as shown in FIGS. 12 and 13. SE and R² are the standard error andcorrelation coefficient, respectively.Symmetric Tree Model. Equation #16 is also validated in symmetric treesfor various organs and species, as shown in FIG. 14. FIG. 14 shows arelation between

$\left( \frac{D_{s}}{D_{\; {m\; {ax}}}} \right)^{\frac{2}{3}}\left( \frac{L_{c}}{L_{m\; {ax}}} \right)$

and normalized crown volume in the symmetric vascular tree for variousorgans and species (21-33), corresponding to the table shown in FIG. 10.Parameters B and A are listed in the table shown in FIG. 10, which havea mean±SD value of 1.02±0.02 and 1.00±0.01, respectively, by averagingover various organs and species. These parameters are in agreement withthe theoretical value of one. Furthermore, Equation #15 implies that

${K_{v} = \frac{V_{m\; {ax}}}{D_{m\; {ax}}^{2/3}L_{m\; {ax}}}},$

which can be compared with the regression-derived value. For the tableshown in FIG. 10, parameters B (obtained from least-square fits) and A(obtained from nonlinear regression with B constrained to one) invarious organs when Equation #16 is represented by Equation #17, whereindependent variables

${X = {{\left( \frac{D_{s}}{D_{m\; {ax}}} \right)^{\frac{2}{3}}\left( \frac{L_{c}}{L_{m\; {ax}}} \right)\mspace{14mu} {and}\mspace{14mu} Y} = \left( \frac{V_{c}}{V_{m\; {ax}}} \right)}},$

as shown in FIG. 14. SE and R² are the standard error and correlationcoefficient, respectively.

FIG. 15 shows a comparison of (K_(v))_(ML) obtained from the nonlinearregression of anatomical data and (K_(v))_(EQ) calculated from Equations#15 and #16. A least-square fit results in a relation of the form:(K_(v))_(EQ)=0.998(K_(v))_(ML) (R²=0.999).

Scaling Relations. To further validate the novel volume scaling law ofthe disclosure of the present application, a number of scaling relationsbetween morphological and hemodynamic parameters are provided below. Forthese relations, parameter A has the theoretical value of one asexponent B has a theoretical value of 3/7, 1 2/7, 2⅓, and 3 fordiameter-length relation, volume-length relation, flow-diameterrelation, and volume-diameter relation in Equations #29-32,respectively. The values for A are listed in the table shown in FIG. 11as determined from nonlinear regression. These values, averaged overvarious organs and species, have mean±SD values of 1.01±0.07, 1.00±0.02,0.99±0.05, and 0.99±0.03 for Equations #29-32, respectively. Theagreement of data with theoretical predictions is excellent asdemonstrated by the data referenced herein. For the table shown in FIG.11, the parameter A obtained from nonlinear regression in various organswhen Equations #29-32 (diameter-length, volume-length, flow-diameter,and volume-diameter relations, respectively) are represented by Equation#17. The exponent B is constrained to 3/7, 1 2/7, 2⅓, and 3 forEquations #29-32, respectively. SE and R² are the standard error andcorrelation coefficient, respectively.

Volume Scaling Law. Many structural and functional features are found tohave a power-law (scaling) relation to body size, metabolic rates, etc.Previous studies showed several scaling relations connecting structurewith function. A novel volume scaling relation of the disclosure of thepresent application has been demonstrated and validated, which relatesthe crown volume to the stem diameter and crown length.

Clinical techniques (e.g., indicator and dye-dilution method) have beenused to predict blood volume for decades. The blood volume variessignificantly with body size such that it is difficult to evaluate thechange of blood volume in patients because of lack of reference.Although Feldschuh and Enson (Prediction of the normal blood volume:relation of blood volume to body habitus. Circulation. 56: 605-612(1977) used the metropolitan life height and weight tables to determinean ideal weight as an approximate reference, this approach lacks aphysical or physiological basis for calculating normal blood volume. Thenovel volume scaling law of the disclosure of the present applicationmay establish the signature of “normality” and deviation thereof may beindicative of pathology.

The remodeling of intravascular volume may be physiologic during normalgrowth, exercise, or pregnancy. It may also be pathological, however, inhypertension, tumor, or diffuse vascular diseases. Diffuse vasculardisease is difficult to quantify because the normal reference does notexist. The disclosure of the present application shows that the volumescaling law holds in the coronary epicardial trees (vessel diameter >1mm), as shown in FIG. 13 and the table shown in FIG. 9. Such data oncoronaries or other vascular trees are available, for example, byangiography, CT, or MRI. Hence, the novel volume scaling law of thedisclosure of the present application can serve to quantify diffusevascular disease in various organs clinically.

Comparison with ZKM Model. As referenced herein, vascular trees providethe channels to transport fluid to different organs. The optimal designof vascular tree is required to minimize energy losses. Although manytheoretical approaches are proposed to explain the design of vasculartree, the “Minimum Energy Hypothesis” may be the most validatedhypothesis. The ZKM model, based on the minimum energy hypothesis,predicted the exponents

${\chi = \frac{{3ɛ^{\prime}} - 2}{4\left( {ɛ^{\prime} + 1} \right)}},{\beta = \frac{5}{ɛ^{\prime} + 1}},{\delta = \frac{4\left( {ɛ^{\prime} + 1} \right)}{{3ɛ^{\prime}} - 2}}$

for diameter-length, volume-length, and flow-diameter relations,respectively, where the parameter ε′ in the exponents is the ratio ofmaximum metabolic to viscous power dissipation for a given tree. Basedon Equations #15 and #16 of the disclosure of the present application,the corresponding exponents

${\chi = \frac{3}{7}},{\beta = {1\; \frac{2}{7}}},{{{and}\mspace{14mu} \delta} = {2\; \frac{1}{3}}}$

are shown. With the respective ε′, the mean values over all organs andspecies are 0.43±0.02, 1.28±0.09, and 2.33±0.11 for exponents χ, β, δ,respectively, which agrees well with the present predicted information,i.e., 3/7≈0.43, 1 2/7≈1.29, and 2⅓≈2.33. Furthermore, ZKM model showsthe mean±SD value of 2.98±0.34 for volume-diameter relation with therespective ε′, which is consistent with the exponent value of 3 inEquation #32. This provides further validation for the proposed volumescaling law of the disclosure of the present application.

Comparison with ¾-power Law. West et al. (A general model for the originof allometric scaling laws in biology. Science. 276:122-126 (1997))proposed the ¾-power scaling law (WBE model) to describe how essentialmaterials are transported in the vascular tree. The WBE model predictsthe following scaling relations: Q_(s)∝M^(3/4), V_(c)∝M, andD_(s)∝M^(3/8). If the first and third relations are combined, oneobtains the flow-diameter relation with an exponent of 8=2, whichimplies that the flow velocity is constant from the large artery to thesmallest arterioles. This is in contradiction with experimentalmeasurements.

If the second and third relations are combined, one obtains thevolume-diameter relation as

${\left( \frac{V_{c}}{V_{m\; {ax}}} \right) = {\left( \frac{D_{s}}{D_{m\; {ax}}} \right)^{\frac{8}{3}} = \left( \frac{A_{s}}{A_{m\; {ax}}} \right)^{\frac{4}{3}}}},$

such that the area-volume relation is

${\left( \frac{A_{s}}{A_{m\; {ax}}} \right) = \left( \frac{V_{c}}{V_{m\; {ax}}\;} \right)^{\frac{3}{4}}},$

where A_(s) and A_(max) are the stem area and the most proximal area,respectively. These WBE predictions differ from the experimentalobservation

$\left( \frac{A_{s}}{A_{m\; {ax}}} \right) = {\left( \frac{V_{c}}{V_{m\; {ax}}\;} \right)^{\frac{2}{3}}.}$

When the cost function in Equation #22 is minimized, one obtains theexponent

${\delta = {2\; \frac{1}{3}}},$

which agrees well with the anatomical data (as shown in the table ofFIG. 10). The area-volume relation

$\left( {\left( \frac{A_{s}}{A_{m\; {ax}}} \right) = \left( \frac{V_{c}}{V_{m\; {ax}}} \right)^{\frac{2}{3}}} \right)$

obtained from Equation #32 is consistent with the experimentalmeasurements.

There is additional departure of the present model from that of WBE.Equation #30 and V_(c)∝M lead to the following relation:

L _(c) ∝M ^(7/9)  (18)

From Equations #18 and #25, the following relation may be identified:

Q _(s) ∝M ^(7/9)  (19)

From Equation #32 and V_(c)∝M, the following relation may be identified:

D _(s) ∝M ^(1/3)  (20)

Although these scaling relations are different from the WBE model,V_(c)∝D_(s) ^(2/3)L_(c) (Equations #18 and #20 and V_(c)∝M) is stillobtained, which further supports the validity of Equations #15 and #16.Equation #19 implies that the ¾-power scaling law (Q_(s)∝M^(3/4=0.75))should be 7/9-power scaling law (Q_(s)∝M^(7/9=0.78)). A least-square fitof Q_(s)−M data has an exponent value of 0.78 (R²=0.985), which isconsistent with the 7/9-power scaling law.

Optimal Cost Function. From Equations #26 and #28, the non-dimensionalcost function can be written as follows:

$\begin{matrix}{f_{c} = {{\frac{1}{6}\frac{\left( {L_{c}/L_{m\; {ax}}} \right)^{3}}{\left( {D_{s}/D_{m\; {ax}}} \right)^{4}}} + {\left( \frac{D_{s}}{D_{m\; {ax}}} \right)^{2/3}\left( \frac{L_{c}}{L_{m\; {ax}}} \right)}}} & (21)\end{matrix}$

This is the minimum cost of maintaining an optimal design of a vasculartree under homeostasis. From the structure-function scaling relations(Equation #29),

${\frac{\left( {L_{c}/L_{m\; {ax}}} \right)^{3}}{\left( {D_{s}/D_{m\; {ax}}} \right)^{4}} = {{\left( \frac{L_{c}}{L_{m\; {ax}}} \right)^{1\; \frac{2}{7}}\mspace{14mu} {{and}\left( \frac{D_{s}}{D_{m\; {ax}}} \right)}^{2/3}\left( \frac{L_{c}}{L_{m\; {ax}}} \right)} = \left( \frac{L_{c}}{L_{m\; {ax}}} \right)^{1\; \frac{2}{7}}}},$

one may obtain

$\frac{\left( {L_{c}/L_{m\; {ax}}} \right)^{3}}{\left( {D_{s}/D_{m\; {ax}}} \right)^{4}} = {\left( \frac{D_{s}}{D_{m\; {ax}}} \right)^{2/3}{\left( \frac{L_{c}}{L_{m\; {ax}}} \right).}}$

The power required to overcome the viscous drag of blood flow (secondterm in Equation #21) is one sixth of the power required to maintain thevolume of blood (third term in Equation #21). This expression impliesthat most of energy is dissipated for maintaining the metabolic cost ofblood, which is proportional to the metabolic dissipation.

Additional Validation of Volume Scaling Law. From Equations #15 and 16,the disclosure of the present application identifies the cost functionfor a crown, F_(c), consistent with previous formulation:

F _(c) =Q _(s) ·ΔP _(c) +K _(m) V _(c) =Q _(s) ² ·R _(c) +K _(m) K _(v)D _(s) ^(2/3) L _(c)  (22)

where Q_(s) and ΔP_(c)=Q_(s)·R_(c) are the flow rate through the stemand the pressure drop in the distal crown, respectively, and K_(m) is ametabolic constant of blood in a crown. The resistance of a crown hasbeen identified as

${R_{c} = {K_{c}\frac{L_{c}}{D_{s}^{4}}}},$

where K_(c) is a constant. The cost function of a crown tree in Equation#22 can be written as:

$\begin{matrix}\begin{matrix}{F_{c} = {{Q_{s}^{2} \cdot R_{c}} + {K_{m}K_{v}D_{s}^{2/3}L_{c}}}} \\{= {{K_{c}Q_{s}^{2}\frac{L_{c}}{D_{s}^{4}}} + {K_{m}K_{v}D_{s}^{2/3}L_{c}}}}\end{matrix} & (23)\end{matrix}$

Equation #23 can be normalized by the metabolic power requirements ofthe entire tree of interest, K_(m)V_(max)=K_(m)K_(v)D_(max)^(2/3)L_(max), to obtain:

$\begin{matrix}\begin{matrix}{f_{c} = {\frac{F_{c}}{K_{m}K_{v}D_{m\; {ax}}^{2/3}L_{m\; {ax}}} =}} \\{= {\frac{Q_{m\; {ax}}^{2}R_{m\; {ax}}}{K_{m}K_{v}D_{m\; {ax}}^{2/3}L_{m\; {ax}}}{\left( \frac{Q_{s}}{Q_{m\; {ax}}} \right)^{2} \cdot}}} \\{{\frac{\left( {L_{c}/L_{m\; {ax}}} \right)}{\left( {D_{s}/D_{m\; {ax}}} \right)^{4}} + {\left( \frac{D_{s}}{D_{m\; {ax}}} \right)^{2/3}\left( \frac{L_{c}}{L_{m\; {ax}}} \right)}}}\end{matrix} & (24)\end{matrix}$

where f_(c), is the non-dimensional cost function. A previous analysisshows:

$\begin{matrix}{Q_{s} = {\left. {K_{Q}L_{c}}\Rightarrow\frac{Q_{s}}{Q_{m\; {ax}}} \right. = \frac{L_{c}}{L_{m\; {ax}}}}} & (25)\end{matrix}$

where K_(Q) is a flow-crown length constant. When Equation #25 isapplied to Equation #24, the dimensionless cost function can be writtenas:

$\begin{matrix}{f_{c} = {{\frac{Q_{m\; {ax}}^{2}R_{m\; {ax}}}{K_{m}K_{v}D_{m\; {ax}}^{2/3}L_{m\; {ax}}} \cdot \frac{\left( {L_{c}/L_{m\; {ax}}} \right)^{3}}{\left( {D_{s}/D_{m\; {ax}}} \right)^{4}}} + {\left( \frac{D_{s}}{D_{m\; {ax}}} \right)^{2/3}\left( \frac{L_{c}}{L_{m\; {ax}}} \right)}}} & (26)\end{matrix}$

Similar to Murray's law, the cost function may be minimized with respectto diameter at a fixed L_(c)/L_(max) to obtain the following:

$\begin{matrix}{\left. \begin{matrix}{\frac{\partial f_{c}}{\partial\left( \frac{D_{s}}{D_{m\; {ax}}} \right)} = \left. 0\Rightarrow{\frac{\left( {- 4} \right)Q_{m\; {ax}}^{2}R_{m\; {ax}}}{K_{m}K_{v}D_{m\; {ax}}^{2/3}L_{m\; {ax}}} \cdot \frac{\left( {L_{c}/L_{m\; {ax}}} \right)^{3}}{\left( {D_{s}/D_{m\; {ax}}} \right)^{5}}} \right.} \\{= {{- \left( \frac{2}{3} \right)}\left( \frac{D_{s}}{D_{m\; {ax}}} \right)^{\frac{2}{3}1}\left( \frac{L_{c}}{L_{m\; {ax}}} \right)}}\end{matrix}\Rightarrow{\frac{6Q_{m\; {ax}}^{2}R_{m\; {ax}}}{K_{m}K_{v}D_{m\; {ax}}^{2/3}L_{m\; {ax}}} \cdot \left( \frac{L_{c}}{L_{m\; {ax}}} \right)^{2}} \right. = \left( \frac{D_{s}}{D_{m\; {ax}}} \right)^{4 + \frac{2}{3}}} & (27)\end{matrix}$

Equation #27 applies to any stem-crown unit. When L_(c)=L_(max) andD_(s)=D_(max) in Equation #27, one may obtain:

$\begin{matrix}{\frac{6Q_{m\; {ax}}^{2}R_{m\; {ax}}}{K_{m}K_{v}D_{m\; {ax}}^{2/3}L_{m\; {ax}}} = {\left. 1\Rightarrow\frac{Q_{m\; {ax}}^{2}R_{m\; {ax}}}{K_{m}K_{v}D_{m\; {ax}}^{2/3}L_{m\; {ax}}} \right. = \frac{1}{6}}} & (28)\end{matrix}$

Therefore, Equation #28 can be written as:

$\begin{matrix}{\left( \frac{D_{s}}{D_{m\; {ax}}} \right) = \left( \frac{L_{c}}{L_{\; {m\; {ax}}}} \right)^{\frac{3}{7}}} & (29)\end{matrix}$

From Equations #16 and #29, one may obtain:

$\begin{matrix}{\left( \frac{V_{c}}{V_{m\; {ax}}} \right) = \left( \frac{L_{c}}{L_{m\; {ax}}} \right)^{1\; \frac{2}{7}}} & (30)\end{matrix}$

From Equations #25 and #29, one may find:

$\begin{matrix}{\left( \frac{Q_{s}}{Q_{m\; {ax}}} \right) = \left( \frac{D_{s}}{D_{m\; {ax}}} \right)^{2\; \frac{1}{3}}} & (31)\end{matrix}$

where Q_(max) is the flow rate through the most proximal stem. FromEquations #29 and #30, one may obtain:

$\begin{matrix}{\left( \frac{V_{c}}{V_{m\; {ax}}} \right) = \left( \frac{D_{s}}{D_{m\; {ax}}} \right)^{3}} & (32)\end{matrix}$

Equations #29-32 are the structure-function scaling relations in thevascular tree, based on the “Minimum Energy Hypothesis”. Equations #29,#30, and #32 represent the diameter-length, volume-length, andvolume-diameter relations, respectively and Equation #31 represents thegeneral Murray's law in the entire tree.

The disclosure of the present application also relates to the design andfabrication of micro-fluidic chambers for use in research anddevelopment, thereby designing a chamber that maximizes flow conditionswhile minimizing the amount of material needed to construct the chamber.

The disclosure of the present application further provides an index thatrelates the myocardial region at risk for infarct and the associatedside branch (SB). This relation can help guide the decision forbifurcation stenting where the SB may be at risk for occlusion. As it isclinically known, the larger the diameter of the SB, the more myocardialmass is at risk. The present disclosure formulates this notionquantitatively and provides a relation where the myocardial region atrisk for infarct can be determined directly from the angiographiccaliber of a SB. This provides a basis for future human studies that maydemarcate the caliber (and hence myocardial mass) in relation to cardiacbiomarkers as a cut off for treatment of SB. Many other uses are alsopossible and within the scope of the disclosure of the presentapplication.

Based on experimental measurements of the cross sectional area of avessel to the perfused myocardial mass, the present disclosure definesthe fraction of myocardial mass perfused by a SB that is at risk forinfarct, relative to the mass perfused by the most proximal artery (%Infarct_(artery)), in the porcine model as shown in Equation #33:

% Infarct_(artery) =M _(SB) /M _(MP)×100=(A _(SB) /A _(MP))^(1/α)×100=(A_(SB) /A _(MP))^(4/3)×100  (33)

In Equation #33, M_(MP) and A_(MP) represent the mass and area of themost proximal main artery and M_(SB) and A_(SB) correspond to therespective SB. Experimental testing found that A˜M^(α) or A_(SB)˜M_(SB)^(α) where α=¾, which is consistent with allometric scaling laws.

FIG. 16A shows a relationship between the SB diameter and the percentageof myocardial mass at risk for infarct relative to the mass perfused bythe most proximal artery. The relation of % Infarct and SB diameter isnon-linear. FIG. 16B shows a relationship between the cross sectionalarea of a side branch and the percentage of myocardial mass at risk forinfarct relative to the mass perfused by the most proximal artery. Therelationship between SB area and % Infarct in FIG. 16B is nearly linear(% Infarct=4.43A_(SB)-4.91, R²=0.994). The data shown in FIGS. 16A and16B are based on the assumption that the diameter of the most proximalartery is 5 mm. The linear relationship between SB area and % Infarctallows the calculation of the potential % Infarct (fraction of entireartery perfused mass) from only the area of the most proximal artery(e.g., left anterior descending artery, left circumflex artery and rightcoronary artery; LAD, LCx, or RCA, respectively) and the lumen area ofthe SB angiographically.

The fraction of myocardial mass perfused by a SB that is at risk forinfarct, relative to the entire heart, may be calculated as shown inEquation #34:

$\begin{matrix}\begin{matrix}{{\% {Infarct}_{heart}} = {\left\lbrack {M_{{SB},P}/\left( {M_{MLMCA} + M_{MRCA}} \right)} \right\rbrack \times 100}} \\{= {\left\lbrack {A_{SB}^{4/3}/\left( {A_{MLMCA}^{4/3} + A_{MRCA}^{4/3}} \right)} \right\rbrack \times 100}}\end{matrix} & (34)\end{matrix}$

wherein LMCA and RCA stand for left main coronary artery and rightcoronary artery, respectively. Hence, if the cross sectional area of theSB of interest as well as the cross sectional area of the main branchesare known, one can determine the fraction of myocardial tissue at riskfor infarct if the SB is occluded.

The % Infarct for the entire heart, as demonstrated in Equation #34, isa smaller percentage than the % Infarct for the most proximal artery,but the shapes of the curves exemplified in FIGS. 16A and 16B remain thesame for the % Infarct_(heart). Assuming that both the LMCA and RCA are5 mm in diameter, one can obtain identical curves except that the %Infarct_(heart)=½% Infarct_(artery) as can be shown by Equations #33 and#34. The relation of % Infarct_(heart) with area is much more linearthan that with diameter. Hence, it is simpler to use the area of the SBrelative to the main branch to estimate the potential myocardial infarctmass.

The above analysis for % Infarct_(heart) and % Infarct_(artery) assumesthat the side branch is normal (i.e., non-diseased). If the SB isdiseased, the diameter or area of the SB would underestimate theperfused myocardial mass. In such case, the diameter of the SB can bedetermined from the mother and other daughter vessel according to the7/3 power law and accordingly used to determined the myocardial mass atrisk.

While various systems and methods to obtain a myocardial mass indexindicative of an at-risk myocardial region have been described inconsiderable detail herein, the embodiments are merely offered by way ofnon-limiting examples of the disclosure described herein. It willtherefore be understood that various changes and modifications may bemade, and equivalents may be substituted for elements thereof, withoutdeparting from the scope of the disclosure. Indeed, this disclosure isnot intended to be exhaustive or to limit the scope of the disclosure.

Further, in describing representative embodiments, the disclosure mayhave presented a method and/or process as a particular sequence ofsteps. However, to the extent that the method or process does not relyon the particular order of steps set forth herein, the method or processshould not be limited to the particular sequence of steps described.Other sequences of steps may be possible. Therefore, the particularorder of the steps disclosed herein should not be construed aslimitations of the present disclosure. In addition, disclosure directedto a method and/or process should not be limited to the performance oftheir steps in the order written. Such sequences may be varied and stillremain within the scope of the present disclosure.

1. A method for diagnosing a risk of cardiac disease, the methodcomprising the steps of: identifying a luminal cross-sectional area of aside branch vessel; identifying a luminal cross-sectional area of a mainartery most proximal to the side branch vessel; determining a myocardialregion at risk for infarct from side branch occlusion relative to a massperfused by the most proximal artery, wherein such region is based on arelationship between the cross-sectional areas of the side branch andthe most proximal main artery; and diagnosing a risk of cardiac diseasebased on a diameter of the side branch vessel and the size of themyocardial region at risk for infarct perfused by the side branch. 2.The method of claim 1, wherein the step of identifying a luminalcross-sectional area of a side branch vessel is performed by coronaryangiography.
 3. The method of claim 1, wherein the step of identifying aluminal cross-sectional area of a main artery most proximal to the sidebranch vessel comprises identifying a luminal cross-sectional area of anartery selected from the group consisting of a left anterior descendingartery, a left circumflex artery, and a right coronary artery.
 4. Themethod of claim 1, wherein the side branch vessel is diseased.
 5. Themethod of claim 1, wherein the step of identifying a luminalcross-sectional area of a side branch vessel is performed using adiameter of a mother vessel of the side branch bifurcation and adiameter of a daughter vessel of the side branch bifurcation.
 6. Amethod for diagnosing a risk of cardiac disease, the method comprisingthe steps of: identifying a luminal cross-sectional area of a sidebranch vessel; identifying a luminal cross-sectional area of a left maincoronary artery; identifying a luminal cross-sectional area of a rightcoronary artery; determining a myocardial region at risk for infarctfrom side branch occlusion relative to a mass perfused by the entireheart, wherein such region is based on a relationship between thecross-sectional areas of the side branch, left main coronary artery andright coronary artery; and diagnosing a risk of cardiac disease based ona diameter of the side branch and the size of the myocardial region atrisk for infarct perfused by the side branch.
 7. The method of claim 6,wherein the step of identifying a luminal cross-sectional area of a sidebranch vessel is performed by coronary angiography.
 8. The method ofclaim 6, wherein the side branch vessel is diseased.
 9. The method ofclaim 8, wherein the step of identifying a luminal cross-sectional areaof a side branch vessel is performed using a diameter of a mother vesselof the side branch bifurcation and a diameter of a daughter vessel ofthe side branch bifurcation.
 10. A system for diagnosing a risk ofcardiac disease, the system comprising: a processor; a storage mediumoperably connected to the processor, the storage medium capable ofreceiving and storing data relative of measurements from a vasculatureof a vessel; wherein the processor is operable to: identify a luminalcross-sectional area of a side branch vessel; identify a luminalcross-sectional area of a main artery most proximal to the side branchvessel; determine a myocardial region at risk for infarct from sidebranch occlusion relative to a mass perfused by the most proximalartery, wherein such region is based on a relationship between thecross-sectional areas of the side branch and the most proximal mainartery; and diagnose a risk of cardiac disease based on a diameter ofthe side branch and the size of the myocardial region at risk forinfarct perfused by the side branch.
 11. The system of claim 10, whereinthe identification of a luminal cross-sectional area of a side branchvessel is performed by coronary angiography.
 12. The system of claim 10,wherein the main artery most proximal to the side branch vesselcomprises the left anterior descending artery.
 13. The system of claim10, wherein the main artery most proximal to the side branch vesselcomprises the left circumflex artery.
 14. The system of claim 10,wherein the main artery most proximal to the side branch vesselcomprises the right coronary artery.
 15. The system of claim 10, whereinthe side branch vessel is diseased.
 16. The system of claim 15, whereinthe identification of a luminal cross-sectional area of a side branchvessel is performed using a diameter of a mother vessel of the sidebranch bifurcation and a diameter of a daughter vessel of the sidebranch bifurcation.
 17. A system for diagnosing a risk of cardiacdisease, the system comprising: a processor; a storage medium operablyconnected to the processor, the storage medium capable of receiving andstoring data relative of measurements from a vasculature of a vessel;wherein the processor is operable to: identify the lumen area of a sidebranch vessel; identify the cross sectional area of the left maincoronary artery; identify the cross sectional area of the right coronaryartery; determine a myocardial region at risk for infarct from sidebranch occlusion relative to a mass perfused by the entire heart,wherein such region is based on a relationship between thecross-sectional areas of the side branch, left main coronary artery andright coronary artery; and diagnose a risk of cardiac disease based on adiameter of the side branch and the size of the myocardial region atrisk for infarct perfused by the side branch.
 18. The system of claim17, wherein the identification of a luminal cross-sectional area of aside branch vessel is performed by coronary angiography.
 19. The systemof claim 17, wherein the side branch vessel is diseased.
 20. The systemof claim 19, wherein the identification of a luminal cross-sectionalarea of a side branch vessel is performed using a diameter of a mothervessel of the side branch bifurcation and a diameter of a daughtervessel of the side branch bifurcation.